On the multiplicity of .
On the number of generators of modules over Laurent Rings.
On the number of generators of modules over polynomial affine rings.
On the Polynomial Ring over a Mori Domain
On the reduction of Kronecker's modular systems whose elements are functions of two and three variables.
On the Regularity of Graded k-Algebras of Krull Dimension ...1.
On the ring of invariants of F*2n.
On the ring of the variety of algebras over a ring
On the torsion free cover of the -module k
On the ultrametric Stone-Weierstrass theorem and Mahler's expansion
We describe an ultrametric version of the Stone-Weierstrass theorem, without any assumption on the residue field. If is a subset of a rank-one valuation domain , we show that the ring of polynomial functions is dense in the ring of continuous functions from to if and only if the topological closure of in the completion of is compact. We then show how to expand continuous functions in sums of polynomials.
On Two Conjectures About Polynomial Rings.
Orderings of monomial ideals
We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular, we give an interpretation of the height function in terms of the Hilbert-Samuel polynomial, and we compute bounds on the maximal order type.
Parametrization of integral values of polynomials
We will recall a recent result about the classification of those polynomial in one variable with rational coefficients whose image over the integer is equal to the image of an integer coefficients polynomial in possibly many variables. These set is polynomially generated over the integers by a family of polynomials whose denominator is and they have a symmetry with respect to a particular axis.We will also give a description of the linear factors of the bivariate separated polynomial over a...
Pascal type properties of Betti numbers.
Pólya fields and Pólya numbers
A number field , with ring of integers , is said to be a Pólya field if the -algebra formed by the integer-valued polynomials on admits a regular basis. In a first part, we focus on fields with degree less than six which are Pólya fields. It is known that a field is a Pólya field if certain characteristic ideals are principal. Analogously to the classical embedding problem, we consider the embedding of in a Pólya field. We give a positive answer to this embedding problem by showing that...
Pólya fields, Pólya groups and Pólya extensions: a question of capitulation
A number field , with ring of integers , is said to be a Pólya field when the -algebra formed by the integer-valued polynomials on admits a regular basis. It is known that such fields are characterized by the fact that some characteristic ideals are principal. Analogously to the classical embedding problem in a number field with class number one, when is not a Pólya field, we are interested in the embedding of in a Pólya field. We study here two notions which can be considered as measures...
Polynomes à valeurs entières et propriété de Skolem.
Polynomes a Valeurs entieres sur un anneau de Pseudovaluation.
Polynomes à valeurs entières sur un anneau non analytiquement irréductible.
Polynômes de Barsky